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2 Automatica (9) 7 77 Contents lists available at ScienceDirect Automatica journal homepage: Technical communique New conditions for delay-derivative-dependent stability, Emilia Fridman, Uri Shaked, Kun Liu School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 9978, Israel a r t i c l e i n f o a b s t r a c t Article history: Received 8 January 9 Received in revised form August 9 Accepted 9 August 9 Available online September 9 Keywords: Time-varying delay Lyapunov Krasovskii functional LMI Two recent Lyapunov-based methods have significantly improved the stability analysis of time-delay systems: the delay-fractioning approach of Gouaisbaut and Peaucelle () for systems with constant delays and the convex analysis of systems with time-varying delays of Park and Ko (7). In this paper we develop a convex optimization approach to stability analysis of linear systems with interval timevarying delay by using the delay partitioning-based Lyapunov Krasovskii Functionals (LKFs). Novel LKFs are introduced with matrices that depend on the time delays. These functionals allow the derivation of stability conditions that depend on both the upper and lower bounds on delay derivatives. 9 Elsevier Ltd. All rights reserved.. Introduction Over the past decades, much effort has been invested in the analysis and design of systems with time delays (see e.g. Fridman & Shaked, ; Hale & Lunel, 99; He, Wang, Lin, & Wu, 7; Kolmanovskii & Myshkis, 999; Niculescu, ; Richard, ). Among the recent advances in this area, two Lyapunovbased methods should be mentioned that significantly improved the stability analysis: the convex analysis of systems with timevarying delays of Park and Ko (7) and the delay-fractioning approach of Gouaisbaut and Peaucelle () for systems with constant delays. These recent methods inspired the present work, where we extend the delay partitioning approach to systems with interval time-varying delay in a convex way. We introduce novel LKFs with matrices that depend on the time delays. This enables us to derive LMI conditions that depend not only on the upper, but also on the lower bound of the delay derivative. The efficiency of the new stability criteria is demonstrated via numerical examples.. Stability of systems with time-varying delay Consider the system ẋ(t) = Ax(t) + A x(t τ(t)), () The material in this paper was presented at ROCOND 9, June 9. This paper was recommended for publication in revised form by Associate Editor Keqin Gu under the direction of Editor André L. Tits. This work was partially supported by Kamea Fund of Israel, by C&M Maus Chair at Tel-Aviv University and by China Scholarship Council. Corresponding author. Tel.: ; fax: addresses: emilia@eng.tau.ac.il (E. Fridman), shaked@eng.tau.ac.il (U. Shaked), liukun@eng.tau.ac.il (K. Liu). -98/$ see front matter 9 Elsevier Ltd. All rights reserved. doi:./j.automatica.9.8. where τ(t) [h a, h b, h a and where A and A are constant matrices. The delay is assumed to be either differentiable with d τ(t) d, () where d and d are given bounds, or fast-varying (with no restrictions on the delay derivative). The initial condition is given by x(t + θ) = φ(θ), θ [ h b,, φ W, where W is the space of absolutely continuous functions φ : [ h b, R n with the square integrable derivative and with the norm φ W = φ() + h b [ φ(s) + φ(s) ds... A delay partitioning approach to stability We divide the delay interval [h a, h b into two segments: [h, h and [h, h, where we denote h = h a, h = h b and h = (h a + h b )/. Then, () can be represented as ẋ(t) = Ax(t) + χ [h,h (τ(t))a x(t τ(t)) + [ χ [h,h (τ(t))a x(t τ(t)), () where χ [h,h : R {, } is the characteristic function of [h, h {, if s [h, h χ [h,h (s) =, otherwise. Consider the following Lyapunov functional: V(t, x t, ẋ t ) = x T (t)p(τ(t))x(t) + + t h x T (s)s x(s)ds + h t h h x T (s)qx(s)ds t τ(t) ξ T S S (s) ξ(s)ds S

3 7 E. Fridman et al. / Automatica (9) (h i+ h i ) i= hi ẋ T (s)r i ẋ(s)dsdθ, () h i+ t+θ where h =, ξ(s) = col{x(s), x(s ( ))}, S S Q, R i >, S >, >. () S We seek P(τ(t)) P(t) of the form [ τ(t) h P(τ(t)) = χ [h,h (τ(t)) [ τ(t) h + [ χ [h,h (τ(t)) i.e. τ(t) h P(τ(t)) = τ(t) h P + h τ(t) P P + h τ(t) P, () P + h τ(t) P, h τ(t) h, P + h τ(t) P, h < τ(t) h, where P k >, k =,,. Note that the function P(t) is continuous in t, since lim τ(t) h P(τ(t)) = P. Following (Hale & Lunel, 99), we define V(t, x t, ẋ t ) = lim sup s + s [V(t + s, x t+s, ẋ t+s ) V(t, x t, ẋ t ). (7) We are seeking for conditions guaranteeing that V α x(t) (8) for some scalar α >. We first consider τ h. We have [ χ(p P P(t) ) ( χ)(p P ) τ h = τ(t) +. (9) Denoting V = x T (t)p(τ(t))x(t), we find [ V τ h = x T (t) P(t)x(t) + ẋ T (t) χ + h τ(t) P + ( χ) [ τ(t) h Moreover, d [ (h i+ h i ) dt i= hi h i+ [ τ(t) h P P + h τ(t) P x(t). () t+θ [ = ẋ T (t) (h i+ h i ) R i ẋ(t) i= (h i+ h i ) i= hi ẋ T (s)r i ẋ(s)dsdθ t h i+ ẋ T (s)r i ẋ(s)ds. () We start with the case of τ [h, h ), where χ =. Using t h j the fact that t h j+ f j (s)ds = τ(t) f t h j (s)ds j+ + h j t τ(t) f j(s)ds,where f j (s) = ẋ T (s)r j ẋ(s), we apply Jensen s inequality (Gu, Kharitonov, & Chen, ) hi [(h i+ h i )f i (s)ds t h i+ hi ẋ T (s)dsr i t h i+ hi ẋ(s)ds, t h i+ hj t τ(t) τ(t) [(h j+ h j )f j (s)ds (τ(t) h j )(h j+ h j )v T j R jv j, t h j+ [(h j+ h j )f j (s)ds (h j+ τ(t))(h j+ h j )v T j R jv j, where v j = τ(t) h j τ(t) hj t τ(t) ẋ(s)ds, v j = h j+ τ(t) t h j+ ẋ(s)ds, () and j =, i =,. Here, for τ(t) h j, we have hj lim ẋ(s)ds = ẋ(t h j ). τ(t) h j τ(t) h j t τ(t) t τ(t) For h j+ τ(t) the vector ẋ(s)ds is defined h j+ τ(t) t h j+ similarly as ẋ(t h j+ ). We thus find V τ<h [ V τ<h + ẋ T (t) (h i+ h i ) R i ẋ(t) i= + x T (t)s x(t) x T (t h )S x(t h ) T x(t h ) S S + x(t h ) x(t h ) S x(t h ) T x(t h ) S S x(t h ) x(t h ) S x(t h ) [x(t) x(t h ) T R [x(t) x(t h ) [x(t h ) x(t h ) T R [x(t h ) x(t h ) (τ(t) h j )(h j+ h j )v T j R jv j (h j+ τ(t))(h j+ h j )v T j R jv j + x T (t h )Qx(t h ) ( τ(t))x T (t τ(t))qx(t τ(t)). () where j =, i =,. We insert free-weighting n n-matrices T, W, Y k, Z k (k =, ) by adding the following expressions to V(t) τ<h : = [x T (t)y T + ẋt (t)y T + xt (t τ(t))t T [ x(t h ) + x(t τ(t)) + (τ(t) h )v, = [x T (t)z T + ẋt (t)z T + xt (t h )W T [x(t h ) + (h τ(t))v x(t τ(t)). () We further use the descriptor method, where the right-hand side of the expression = [x T (t)p T j + ẋt (t)p T j [Ax(t) + A x(t τ(t)) ẋ(t), () with some n n-matrices P j, P j is added into the right-hand side of (). Setting η j (t) = col{x(t), ẋ(t), x(t h ), x(t h ), v j, v j, x(t τ(t)), x(t h )}, we obtain that along () V τ<h η T (t)ψ τ<h η (t) α x(t), () for some scalar α > if Ψ τ<h Ω + τ(t)(p P ) Ω R Y T Z T (τ(t) h )Y T Ω Y T Z T (τ(t) h )Y T φ () S φ () φ ()

4 E. Fridman et al. / Automatica (9) (h τ(t))z T Ω 7 (h τ(t))z T Ω 7 T R + W S <, (7) (τ(t) h )T φ () (h τ(t))w ( τ(t))q + T + T T W (S + R ) where Ω j = AT P j + P T j A + S R, [ τ(t) Ω = h P + h τ(t) P P T h h + AT P, Ω j = P j P T + j (h i+ h i ) R i, i= Ω j 7 = Y T j Z T j + P T j A, Ω j 7 = Y T j Z T j + P T j A, φ () = (S + R S Q ), φ () = (S + R S ), φ () = ( )(τ(t) h )R, φ () = ( )(h τ(t))r, i, j =,. (8) The latter LMI leads for τ(t) h and for τ(t) h to the following LMIs: Ω + τ(t)(p P ) Ω R τ(t)=h Y T Z T Ω Y T Z T φ () Ψ = S φ () ( )Z T Ω 7 ( )Z T Ω 7 T R + W S <, (9) φ () (h τ(t)=h h )W ( τ(t))q + T + T T W (S + R ) and Ω + τ(t)(p P ) Ψ = Ω τ(t)=h R Y T Z T Ω Y T Z T φ () S φ () ( )Y T Ω 7 ( )Y T Ω 7 T R + W S <. () φ () (h τ(t)=h h )T ( τ(t))q + T + T T W (S + R ) It is easy to see that Ψ results from Ψ τ<h,τ=h, where we have deleted the zero row and the zero column. Denoting: η i (t) = col{x(t), ẋ(t), x(t h ), x(t h ), v ji, x(t τ(t)), x(t h )}, i =,, the latter two LMIs imply () because h τ(t) η T h h (t)ψ η (t) + τ(t) h η T h h (t)ψ η (t) = η T (t)ψ τ<h η (t) α x(t) and Ψ τ<h is thus convex in τ(t) [h, h ). LMI (9) leads for τ(t) = d i, i =, to the following: Ψ i = Ψ τ(t)=di <, i =,. () The two LMIs () imply (9) because d τ(t) d d Ψ + τ(t) d d d Ψ = Ψ <. and Ψ is thus convex in τ(t) [d, d. Similarly, we can obtain that Ψ is also convex in τ(t) [d, d. For τ(t) (h, h, where χ =, we apply the above arguments and representations with j = and i =,. Similarly to (), we insert free-weighting n n-matrices T, W, Y k, Z k (k =, ) by adding the following expressions to V : = [x T (t)y T + ẋt (t)y T + xt (t τ(t))t T + xt (t h )W T [ x(t h ) + x(t τ(t)) + (τ(t) h )v, = [x T (t)z T + ẋt (t)z T [x(t h ) + (h τ(t))v x(t τ(t)). Similarly to (), the expression with j = is added to V. We then arrive at the following: V τ>h η T (t)ψ τ>h η (t), where Ψ τ>h Ω + τ(t)(p P ) Ω R Y T (τ(t) h )Y T Ω Y T (τ(t) h )Y T φ () R W T + S (τ(t) h )W T φ () φ () (h τ(t))z T Ω 7 Z T (h τ(t))z T Ω 7 Z T W T T S <, () (τ(t) h )T φ () ( τ(t))q + T + T T S and where [ τ(t) Ω = h P + h τ(t) P φ () = (S + R + R S Q ), φ () = (S + R S ), φ () = ( )(τ(t) h )R, φ () = ( )(h τ(t))r. P T + AT P, () We note that Ψ τ>h is convex in τ(t) (h, h and, thus, for the feasibility of LMI (), it is sufficient to verify this LMI for τ(t) h and for τ(t) h. Denote the resulting LMIs by

5 7 E. Fridman et al. / Automatica (9) 7 77 Ψ < and Ψ <, respectively, where the zero columns and rows are deleted from Ψ and Ψ. Clearly Ψ and Ψ are also convex in τ(t) [d, d. Along the system (), we therefore have V τ h χ [h,h (τ(t))η T (t)ψ τ<h η (t) + [ χ [h,h (τ(t))η T (t)ψ τ>h η (t) α x(t) () for some positive scalar α >. For τ = h, taking into account the definition (7) of V, we find V τ=h max{η T (t)ψ τ<h η (t), η T (t)ψ τ>h η (t)} α x(t). () By using Theorem 8.. of (Kolmanovskii & Myshkis, 999), we finally obtain the following: Theorem. Let there exist n n-matrices Q, R i (i =,, ), S, S, S, S, satisfying (), and n n-matrices P k >, k =,,, W, W, P j, P j, Y j, Y j, T j, Z j and Z j, j =, such that the eight LMIs: (7), for τ(t) h and τ(t) h, and (), for τ(t) h and τ(t) h, where τ(t) = d, d, with notations given in (8) and (), are feasible. Then () is asymptotically stable for all differentiable delays τ(t) [h, h with d τ(t) d. For unknown d, by substituting P = P = P = P and τ(t) = d into (7) and (), we arrive at the following: Corollary. Let there exist n n-matrices Q, R i (i =,, ), S, S, S, S, satisfying (), and n n-matrices P >, W, W, P j, P j, Y j, Y j, T j, Z j and Z j, j =, such that the four LMIs: (7), for τ(t) h and τ(t) h, and (), for τ(t) h and τ(t) h, where τ(t) = d, with notations given in (8) and (), are feasible. Then () is asymptotically stable for all differentiable delays τ(t) [h, h with τ(t) d. Moreover, if the above LMIs are feasible with Q =, then () is asymptotically stable for all fastvarying delays in [h, h... On other possibilities for delay partitioning If h a is big enough, delay partitioning of [, h a can improve the result. For simplicity we combine delay partitioning of [, h a with a non-partitioned [h a, h b, where h a = h and h b = h. We apply V(t, x t, ẋ t ) = x T (t)p(τ(t))x(t) x(s) ( + t h x s h ) S S T ( x(s) S x s h ) ds + h ẋ T (s)r ẋ(s)dsdθ h t+θ h h + x T (s)s x(s)ds + x T (s)qx(s)ds t h t τ(t) h + ( ) ẋ T (s)r ẋ(s)dsdθ, () h t+θ where τ(t) [h, h and where P(τ(t)) = τ(t) h P + h τ(t) P, P >, P >, (7) S S Q, R >, R >, S >, >. (8) S We have: d dt [xt (t)p(τ(t))x(t) = x T (t) τ(t)(p P ) x(t) [ τ(t) + ẋ T h (t) P + h τ(t) P x(t). By using arguments of Theorem (without partitioning of [h a, h b ), we obtain that for some α > V η T (t)φη(t) α x(t), (9) where η(t) = col{x(t), ẋ(t), x(t h ), x(t h ), v, v, x(t τ(t)), x(t h )}, if the LMI Φ + τ(t)(p P ) Φ Y T Z T (τ(t) h )Y T Φ Y T Z T (τ(t) h )Y T S S + Q Φ = S Φ (h τ(t))z T Y T Z T + P T A S + R (h τ(t))z T Y T Z T + P T A T S T (τ(t) h )T Φ ( τ(t))q + T + T T S S R holds, where Φ = A T P + P T A + S R, [ τ(t) h Φ = P + h τ(t) P P T h h + AT P, Φ = P P T + h R + ( ) R, Φ = ( )(τ(t) h )R, Φ = ( )(h τ(t))r. The following result is then obtained. < () () Theorem. Let there exist n n-matrices Q, R, S, S, S, R, S, satisfying (8), and n n-matrices P >, P >, P, P T, Y, Y, Z, Z such that the four LMIs: (), for τ(t) h and τ(t) h, where τ(t) = d, d, with notations given in (), are feasible. Then () is asymptotically stable for all differentiable delays h τ(t) h satisfying d τ(t) d. When d is unknown, by setting P = P = P, τ(t) = d in (), we obtain the following Corollary. Let there exist n n-matrices Q, R, S, S, S, R, S, satisfying (8), and n n-matrices P >, P, P T, Y, Y, Z, Z such that the two LMIs: (), for τ(t) h and τ(t) h, where τ(t) = d, with notations given in (), are feasible. Then () is asymptotically stable for all differentiable delays h τ(t) h satisfying τ(t) d. Moreover, if the above LMIs are feasible with Q =, then () is asymptotically stable for all fast-varying delays in [h, h. Remark. The examples below illustrate that for big enough h a (for big enough h b h a ) the delay partitioning of [, h a (of [h a, h b ) improve the result. Thus, in Example, the results by Corollary are worse for h a and better for h a than the ones by Corollary. Partitioning of the above intervals into n > subintervals may lead to further improvements. In (Jiang & Han, 8) another delay partitioning was introduced, which corresponded to the partitioning into two subintervals of [, h a and of [, h b. Example below shows that our approach yields less conservative results.

6 E. Fridman et al. / Automatica (9) Table Example : Max. value of h b achieved for h a =. d \ d....7 unkn. Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Theorem Table Example : Max. value of h b for h a =. d \ Method He7 Shao9 Corollary Corollary..... unknown Table Example : Max. value of h b achieved for h a =. d \ d.... Theorem Theorem Theorem Theorem Examples Example. Consider () with A = and A =. For h a =, choosing d and d as in Table and applying Theorems and, we find the maximum values of h b for which the system remains asymptotically stable (see Table ). For unknown d, our results coincide with those of the corresponding Corollaries and. The result of Corollary coincides with the one of Park and Ko (7) (the latter are less conservative than those of He et al. (7)and Shao (9)), but the LMIs of Corollary possess a fewer number of decision variables. It is seen that the value of h b depends on d and it grows for d. For h a =, d unknown and d =. or unknown, the results obtained by various methods in the literature for the admissible upper-bounds h b, which guarantee the stability of the system () are listed in Table. For h a =, choosing d and d and applying Theorems and, we obtain the results given in Table. Example. Consider () with A = and A.9 =. For h a = we find, by applying Corollary, that the system is asymptotically stable for all fast-varying delays τ(t) [,.88, which coincides with the result of Park and Ko (7) (the latter is less conservative than those of He (7), Jiang and Han (8) and Shao (9)). Corollary leads to a bigger interval τ(t) [,.8. Table Example : Max. value of h b for different h a. h a Jiang and Han (8) Shao (9) Corollary Corollary Table Example : Max. value of h b achieved for d =. h a \ d..7 unknown Theorem Theorem For h a =,,,, and fast-varying delays we obtain, by applying Corollaries and, the maximum values of h b given in Table. These results are favorably compared with the existing ones. For d =, choosing d and h a as in Table we apply Theorems and. The results by Theorem are d -dependent (see Table ), where the value of h b grows for d. For unknown d the results coincide with those of Corollary. The results by Theorem are d -independent and coincide with the ones by Corollary.. Conclusions In this paper new LKFs with matrices depending on the time delay are introduced. These delay partitioning-based LKFs lead to stability conditions that depend on both the upper and lower bounds on the delay derivative. Two examples illustrate the efficiency of the new method and the improvement that can be achieved by using the lower bound on the delay derivative. Acknowledgements We thank the associate editor and the reviewers for their suggestions, which have improved the quality of the paper. References Fridman, E., & Shaked, U. (). A descriptor approach to H control of linear time-delay systems. Institute of Electrical and Electronics Engineers. Transactions on Automatic control, 7(), 7. Gouaisbaut, F., & Peaucelle, D. (). Delay-dependent stability of time delay systems. In: Proc. of ROCOND, oulouse. Gu, K., Kharitonov, V., & Chen, J. (). Stability of time-delay systems. Boston: Birkhauser. Hale, J., & Lunel, S. (99). Introduction to functional differential equations. New York: Springer-Verlag. He, Y., Wang, Q.-G., Lin, C., & Wu, M. (7). Delay-range-dependent stability for systems with time-varying delay. Automatica,, 7 7. Jiang, X., & Han, Q.-L. (8). New stability criteria for linear systems with interval time-varying delay. Automatica,, 8 8. Kolmanovskii, V., & Myshkis, A. (999). Applied theory of functional differential equations. Kluwer. Niculescu, S. I. (). Lecture Notes in Contr. and Inf. Sciences: Vol. 9. Delay effects on stability: A Robust Control Approach. London: Springer-Verlag. Park, P., & Ko, J. W. (7). Stability and robust stability for systems with a time-varying delay. Automatica,, Richard, J.-P. (). Time-delay systems: An overview of some recent advances and open problems. Automatica, 9, 7 9. Shao, H. (9). New delay-dependent criteria for systems with interval delay. Automatica,, 7 79.

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